According to this quote, Kefalonia is exempt from paying the property tax for 2014, BUT only those property owners who are full residents. This is a Google translation of the report.

Quote

This setting applies only to residents and businesses Kefalonia island, not all those who have property in Kefalonia - decided and discharge of individual real estate properties that are completely unusable due to past earthquakes in other regions of Greece The discharge of the whole of Kefalonia ENFIA decided during the meeting of Prime Minister Antonis Samaras, the Deputy Prime Minister, Evangelos Venizelos, the finance minister, Gikas Hardouvelis at the Maximos Mansion.

This government commitment after the recent earthquake.

As stated late Tuesday afternoon by the press office of the Ministry of Finance, the exemption of Kefalonia from ENFIA for this year only residents and businesses operating on the island. In evnoii setting excludes those who have owned the island but are not permanent residents of Kefalonia.

In parallel, the government meeting under Prime Minister decided, while the discharge of individual real estate properties that are completely unusable due to past earthquakes in other parts of Greece (as in the cases was relocating residents).

The post office (ELTA) issued a statement on Monday saying that 59 of its outlets would stay open later than usual on Monday and Tuesday to allow taxpayers to pay the first instalment of the unified property tax (ENFIA).

There were long queues at banks and post offices on Monday as Greeks sought to beat the September 30 deadline for the first ENFIA tranche.

It's astonishing how so few people seem to pay this sort of thing via e-banking and will queue for yonks to do it or even pay an accountant to print the instalments off. Probably an indication of how little the real-life value use of computers and online transactions has permeated the Greek psyche and how used people here are to ταλαιπωρία (hassle) and resigned to waiting in queues.

The Single Property Tax (ENFIA) will be abolished, an amendment on unpaid rents will be tabled and the tax-free threshold will rise to 12,000 euros per year, Alternate Finance Minister Nadia Valavani announced on Thursday, while Minister Yanis Varoufakis stated that “we will see the end” of the solidarity tax.

After its pre-election pledge to raise the minimum salary to 751 euros per month, two more of the government’s promises have now been postponed for 2016: the abolition of the Single Property Tax (ENFIA), which will continue this year as it was in 2014, and the increase in the income tax-free threshold to 12,000 euros per year, which won’t happen until next year.

Prime Minister Alexis Tsipras said in a televised interview late on Monday that “abolishing ENFIA and raising the tax-free threshold depend on the course of the economy and the capacity we will have to meet our targets. Otherwise they will be put off till 2016.”

Given the poor performance of the fiscal figures, it is virtually impossible for ENFIA and the Supplementary Property Tax to be replaced by the Large Property Tax (FMAP) that governing SYRIZA had promised before the January election.

It also remains unknown whether ENFIA will be calculated according to the old objective values (property rates used for tax purposes) or the new ones that are currently being drafted by a Finance Ministry committee.

The Finance Ministry is promising a reduction on the Single Property Tax (ENFIA) by 30 percent by cutting the property rates used for tax purposes, known as “objective values.”

Officials close to Alternate Minister Nadia Valavani said that in such a case revenues from property taxes will drop from the 2.65 billon provided for in the budget to 1.855 billion euros, creating a shortfall of 795 million euros, which will then have to be covered by other means of revenues.

Greece’s proposal to its creditors includes the maintenance of state revenues from property taxation at 2.65 billion euros per year both for 2015 and 2016 – i.e. the same target that was introduced and collected by the previous government in 2014.

This is one of the few measures that the government and the country’s creditors have apparently agreed on, pointing to the Single Property Tax (ENFIA) staying on for at least two years. This means that any reductions to property taxation as cited by government officials will be postponed until 2017 at the earliest.

The government will not only continue to impose the Single Property Tax (ENFIA) on owners but it will also use existing objective values – property rates used for tax purposes that are typically above market prices – to determine the level of the tax payable by owners.

Sources say that the ministry is trying to shift some of the burden from the main to the supplementary tax, which is payable only by owners whose properties have an objective value of at least 300,000 euros.

Greece's Alternate Finance Minister Tryfon Alexiadis on Wednesday announced plans to revise a special property tax with the aim of lightening the burden for low incomes and small property holders, without changing a 2.65 billion euros revenue target. He also announced a package of measures to intensify controls in the market and in tourist regions of the country.

Alexiadis said that the ministry was examining measures exploiting credit cards for direct VAT payments and announced that a plan for an online connection of cash machines with the ministry’s IT authority would begin from tourist islands. He dismissed the possibility of further extending a deadline for income tax statements.

He also announced a package of measures aimed at intensifying tax controls in the market and particularly in tourist regions. He added that inspections carried out over the weekend on a total of 1,600 enterprises found that tax evasion rates were around 23 pct.

Alternate Finance Minister Tryfon Alexiadis... has received instructions to issue the bills for the ENFIA property tax by the end of the month. The 2015 ENFIA will be paid in five installments, the first at the end of October and the last in late February 2016.